C25 Page 473 (paper version of the book)

\[T = \left[\begin{array}{rrr} x_1\\ x_2\\ x_3 \end{array}\right] \left[\begin{array}{rr} 2x_1 - x_2 +5x_3 \\ -4x_1 + 2x_2 - 10x_3 \end{array}\right] \] \[= x_1 \left[\begin{array}{rr} 2\\ -4 \end{array}\right] + x_2 \left[\begin{array}{rr} -1\\ 2 \end{array}\right] + x_3 \left[\begin{array}{rr} 5\\ -10 \end{array}\right] \]

\[ = \left[\begin{array}{rr} 2 & -1 & 5 \\ -4 & 2 & -10 \end{array}\right] \left[\begin{array}{rr3} x_1 \\ x_2 \\ x_3 \end{array}\right] \]

This is a linear transformation by definition of the Matrix Build Proof: